Problem and Main Results
Background/Remarks/Motivations/Open Questions
Existence of extremals — Proof
Properties of Extremals
Extremals For The Tomas-Stein Inequality
M. Christ, UCB
Shuanglin Shao, IMA
October 2009
M. Christ, UCB Shuanglin Shao, IMA
Extremals For The Tomas-Stein Inequality
Problem and Main Results
Background/Remarks/Motivations/Open Questions
Existence of extremals — Proof
Properties of Extremals
Outline
1
Problem and Main Results
2
Background/Remarks/Motivations/Open Questions
3
Existence of extremals — Proof
4
Properties of Extremals
Regularity
Complex-valued extremals
Local Maxima
M. Christ, UCB Shuanglin Shao, IMA
Extremals For The Tomas-Stein Inequality
Problem and Main Results
Background/Remarks/Motivations/Open Questions
Existence of extremals — Proof
Properties of Extremals
The Inequality
kfc
σkL4 (R3 ) ≤ C kf kL2 (S 2 ,σ) .
Define R = optimal constant C .
Extremizer: f 6= 0 such that equality holds with R.
Extremizing sequence {fν }: kfν kL2 (S 2 ) ≤ 1 and kfc
ν σk4 → R.
M. Christ, UCB Shuanglin Shao, IMA
Extremals For The Tomas-Stein Inequality
Problem and Main Results
Background/Remarks/Motivations/Open Questions
Existence of extremals — Proof
Properties of Extremals
Main Results
Extremizers exist.
Any extremizing sequence of nonnegative functions is
precompact in L2 (S 2 ).
Nonnegative extremizers satisfy:
They are even: f (−x) ≡ f (x).
They are C ∞ .
The set of nonnegative extremizers of norm 1 is compact.
M. Christ, UCB Shuanglin Shao, IMA
Extremals For The Tomas-Stein Inequality
Problem and Main Results
Background/Remarks/Motivations/Open Questions
Existence of extremals — Proof
Properties of Extremals
A Group of Symmetries
Multiplying f (x) by e iξ·x changes neither kf k2 nor kfc
σk4 .
In local coordinates, the phase x 7→ x · ξ looks approximately
quadratic, not linear.
Theorems.
Any complex extremizer is of the form e ix·ξ f (x) for some
nonnegative extremizer f and some ξ ∈ R3 .
If {fν } is a complex extremizing sequence then there exists
{ξν } so that {e −ix·ξν fν (x)} is precompact.
M. Christ, UCB Shuanglin Shao, IMA
Extremals For The Tomas-Stein Inequality
Problem and Main Results
Background/Remarks/Motivations/Open Questions
Existence of extremals — Proof
Properties of Extremals
(My) Motivation
A δ quasiextremal for an inequality is a vector v satisfying
kTv kY ≥ δkv kX .
Characterizations/necessary conditions for quasiextremals
refine inequalities.
There are three natural regimes: δ ∼ 1, δ → 0, and
δ −→ optimal constant in the inequality.
Once one has such results in the first two regimes, it is natural
to ask what additional information can be squeezed out in the
third.
M. Christ, UCB Shuanglin Shao, IMA
Extremals For The Tomas-Stein Inequality
Problem and Main Results
Background/Remarks/Motivations/Open Questions
Existence of extremals — Proof
Properties of Extremals
Open Questions
Constant functions are local maxima of the functional
Constant functions extremize
related
L4
7→
L4
kfc
σk44
,
kf k44
kfc
σk44
.
kf k42
and a large family of
inequalities.
Do constant functions extremize our inequality?
Are extremizers unique modulo rotation and multiplication by
ce ix·ξ ?
What is the optimal constant in the inequality?
M. Christ, UCB Shuanglin Shao, IMA
Extremals For The Tomas-Stein Inequality
Problem and Main Results
Background/Remarks/Motivations/Open Questions
Existence of extremals — Proof
Properties of Extremals
Extensions
The main results also hold for b : L2 (S 1 ) → L6 (R2 ).
Work on higher dimensions is underway, but nonnegative
functions don’t play same role; results/proofs must be
somewhat different.
Symmetry of S 2 is not essential. (Currently under
investigation by a student . . . )
M. Christ, UCB Shuanglin Shao, IMA
Extremals For The Tomas-Stein Inequality
Problem and Main Results
Background/Remarks/Motivations/Open Questions
Existence of extremals — Proof
Properties of Extremals
Remarks
Inequality is equivalent to
kf σ ∗ f σkL2 (R3 ) ≤ S 2 kf k2L2 (S 2 )
where S = (2π)−3/4 R. We work with f σ ∗ f σ rather than with fc
σ.
Nonnegative functions play a special role:
kf σ ∗ f σkL2 ≤ k|f |σ ∗ |f |σkL2 .
Results of this type are already known for the paraboloid P2 ,
equipped with the appropriate measure.
Foschi gave a very very simple proof for P2 and P1 : Two successive
applications of Cauchy-Schwarz; both inequalities become equalities
if and only if f is Gaussian.
M. Christ, UCB Shuanglin Shao, IMA
Extremals For The Tomas-Stein Inequality
Problem and Main Results
Background/Remarks/Motivations/Open Questions
Existence of extremals — Proof
Properties of Extremals
Concentration Compactness
Kunze proved existence of extremizers and precompactness of
nonnegative extremizing sequences for the L2 7→ L6 inequality
for P1 , using concentration compactness techniques.
Lions (1984,1985) in a series of seminal papers developed the
method of concentration compactness. He points out that
while the pattern is very general, certain points need to be
worked out in each particular case. We have not relied on any
elements of general theory, but our analysis does follow the
pattern laid out by Lions.
M. Christ, UCB Shuanglin Shao, IMA
Extremals For The Tomas-Stein Inequality
Problem and Main Results
Background/Remarks/Motivations/Open Questions
Existence of extremals — Proof
Properties of Extremals
Ingredients
Concentration compactness
Inequalities for convolutions
Approximate scaling, and comparison with a scaling limit
Fourier integral operators
Symmetrization
A characterization of approximate characters of R3
A nonlocal Euler-Lagrange equation
Regularity theorem for an equation with critical scaling
An idea from additive combinatorics
Spherical harmonics and Gegenbauer polynomials
Several explicit computations
M. Christ, UCB Shuanglin Shao, IMA
Extremals For The Tomas-Stein Inequality
Problem and Main Results
Background/Remarks/Motivations/Open Questions
Existence of extremals — Proof
Properties of Extremals
Two Principles
Suppose that f + g is a (1 − δ)-near extremal, f ⊥ g ,
kf + g k2 = 1.
If kg k2 ≥ ε then g is an η(ε)-quasiextremal,
provided δ ≤ δ(ε).
If min kf k2 , kg k2 ≥ ε then kf σ ∗ g σk2 ≥ η(ε),
provided δ ≤ δ(ε).
Thus bilinear estimates which exploit incompatible structural
properties of f , g are a key.
M. Christ, UCB Shuanglin Shao, IMA
Extremals For The Tomas-Stein Inequality
Problem and Main Results
Background/Remarks/Motivations/Open Questions
Existence of extremals — Proof
Properties of Extremals
Steps
Step 1. The inequality is equivalent to
kf σ ∗ f σk2 ≤ S 2 kf k22 .
Restrict henceforth to nonnegative functions.
Step 2. Let f ∗ denote the L2 symmetrization of f with
respect to x 7→ −x. Then
kf σ ∗ f σk2 ≤ kf ∗ σ ∗ f ∗ σk2
with equality only if f is even. Restrict henceforth to even
nonnegative functions.
Proof of inequality: Concrete extremization problem over a
four-dimensional manifold.
M. Christ, UCB Shuanglin Shao, IMA
Extremals For The Tomas-Stein Inequality
Problem and Main Results
Background/Remarks/Motivations/Open Questions
Existence of extremals — Proof
Properties of Extremals
Limiting Problem
Let {fν } be an extremizing sequence with kfν k2 = 1. A
possibility which must be excluded is that
fν2 * 12 δy + 21 δ−y .
?
Formally, as fν concentrates at y , our inequality converges,
after renormalization, to a limiting problem, which is the same
inequality for the paraboloid
P2 = {(x1 , x2 , x3 ) : x3 = 21 (x12 + x22 )}.
Important: How are best constants S, P related?
M. Christ, UCB Shuanglin Shao, IMA
Extremals For The Tomas-Stein Inequality
Problem and Main Results
Background/Remarks/Motivations/Open Questions
Existence of extremals — Proof
Properties of Extremals
Step 3. S > (3/2)1/4 P. Strict inequality is crucial; its role will be
explained later.
The factor (3/2)1/4 is entirely natural and simply takes into account
the fact that what is relevant is weak convergence to a sum of two
Dirac masses with coefficients 12 , rather than one.
First proof: Simply compute kf σ ∗ f σk2 for f ≡ 1. Deduce
S ≥ 21/4 P.
Alt. proof: Perturbative calculation using trial functions e x·ξ for
ξ ∈ R3 as |ξ| → ∞.
M. Christ, UCB Shuanglin Shao, IMA
Extremals For The Tomas-Stein Inequality
Problem and Main Results
Background/Remarks/Motivations/Open Questions
Existence of extremals — Proof
Properties of Extremals
Step 4
Define caps C(z, r ) ⊂ S 2 with center z ∈ S 2 and radius r > 0.
Lemma. (Moyua, Vargas, and Vega)
For any δ > 0 there exist Cδ < ∞ and ηδ > 0 such that: If
f ∈ L2 (S 2 ) satisfies kf σ ∗ f σk2 ≥ δ 2 S 2 kf k22 then there are an
orthogonal decomposition f = g + h and a cap C satisfying
kg k2 ≥ηδ kf k2
and
|g (x)| ≤ Cδ kf k2 |C|−1/2 χC (x) ∀x.
M. Christ, UCB Shuanglin Shao, IMA
Extremals For The Tomas-Stein Inequality
Problem and Main Results
Background/Remarks/Motivations/Open Questions
Existence of extremals — Proof
Properties of Extremals
Step 5: Uniform Scale-Invariant Bounds
To a cap C(z, r ) is associated a rescaling map which identifies
C with the unit ball in R2 . φ∗C : L2 (S 2 ) → L2 (R2 ) is essentially
unitary.
Defn: f is δ–nearly extremal if kf σ ∗ f σk22 ≥ (1 − δ)S 4 kf k42 .
Proposition. Let f be nonnegative, even, δ–nearly extremal,
kf k2 = 1. Then there are an associated cap C and
decomposition f = F + G with
F , G are even and nonnegative with disjoint supports,
φ∗C (F ) satisfies uniform upper bounds,
kG k2 < ε,
ε(δ) → 0 as δ → 0.
M. Christ, UCB Shuanglin Shao, IMA
Extremals For The Tomas-Stein Inequality
Problem and Main Results
Background/Remarks/Motivations/Open Questions
Existence of extremals — Proof
Properties of Extremals
These bounds include
Higher integrability
Z
x:φ∗C (f )(x)>R
φ∗C (f )(x)
2
dx → 0
with a uniform rate independent of f as R → ∞;
Similar L2 decay as |x| → ∞.
Main ingredient: Elementary bilinear inequality
kf σ ∗ g σk2 kf k2 kg k2
when f , g are characteristic functions of two caps which either
have ratio of radii large, or distance between centers large relative
to radii.
M. Christ, UCB Shuanglin Shao, IMA
Extremals For The Tomas-Stein Inequality
Problem and Main Results
Background/Remarks/Motivations/Open Questions
Existence of extremals — Proof
Properties of Extremals
Step 6: Regularity After Rescaling
Let f be δ–nearly extremal, nonnegative , and even.
Let F = φ∗C (f ) be as before.
Lemma. Let ε > 0. If δ is sufficiently small then F may be
decomposed as Gε + Hε where
kGε kC 1 ≤ C (ε).
kHε k2 < ε,
C (ε) does not depend on f .
M. Christ, UCB Shuanglin Shao, IMA
Extremals For The Tomas-Stein Inequality
Problem and Main Results
Background/Remarks/Motivations/Open Questions
Existence of extremals — Proof
Properties of Extremals
Idea/Substeps
Since F ≥ 0 , kF k2 = 1 and F is approximately localized to
the unit ball,
Z
|Fb (ξ)|2 dξ ≥ c > 0
|ξ|≤1
with c an absolute constant.
If F also has a substantial high-frequency component, then
interactions between high-frequency and low-frequency
components produce negligible cross terms .
This violates second of our two basic principles.
M. Christ, UCB Shuanglin Shao, IMA
Extremals For The Tomas-Stein Inequality
Problem and Main Results
Background/Remarks/Motivations/Open Questions
Existence of extremals — Proof
Properties of Extremals
Step 7: Excluding Small Caps
This is the crucial step in which we finally exclude
fν2 * 21 δx0 + 12 δ−x0 .
Proposition. If {fν } is an extremizing sequence of
nonnegative even functions, then the radii of the associated
caps Cν cannot tend to zero.
Once small caps are excluded, any nonnegative even
extremizing sequence must be precompact by the uniform
estimates.
M. Christ, UCB Shuanglin Shao, IMA
Extremals For The Tomas-Stein Inequality
Problem and Main Results
Background/Remarks/Motivations/Open Questions
Existence of extremals — Proof
Properties of Extremals
Ingredients in Excluding Small Caps
Use corresponding inequality for paraboloid
P2 : x3 = 12 (x12 + x22 ).
The optimal constant for S 2 is strictly larger than for P2 , and
with a factor > (3/2)1/4 which accounts for even functions.
The rescaled functions φ∗ν (fν ) are precompact in L2 (R2 ).
This precompactness is essential in showing that if the radii
rν → 0, then the paraboloid inequality controls the sphere
inequality for fν .
M. Christ, UCB Shuanglin Shao, IMA
Extremals For The Tomas-Stein Inequality
Problem and Main Results
Background/Remarks/Motivations/Open Questions
Existence of extremals — Proof
Properties of Extremals
Regularity
Complex-valued extremals
Local Maxima
Regularity of Critical Points
Euler-Lagrange equation:
f = λΛ(f ) = λ f σ ∗ f σ ∗ f σ 2 .
S
Theorem. All L2 solutions are C ∞ .
f ∈ H s (S 2 ) ⇒ Λ(f ) ∈ H s+δ provided s > 0 , but equation is
critical in H 0 in sense of scaling.
Main step: f ∈ H 0 solution ⇒ f ∈ H s where s = s(λ, f )
depends in principle not only on λ and kf k2 , but also on the
“profile” of f .
Idea: Any particular solution f breaks the scaling symmetry.
The following argument is known in analogous PDE situations.
M. Christ, UCB Shuanglin Shao, IMA
Extremals For The Tomas-Stein Inequality
Problem and Main Results
Background/Remarks/Motivations/Open Questions
Existence of extremals — Proof
Properties of Extremals
Regularity
Complex-valued extremals
Local Maxima
Regularity, continued
Split f = ϕε + bε where ϕε ∈ C ∞ and kbε kH 0 < ε 1.
Rewrite equation: bε = Lε (bε ) + Nε (bε ) where Lε is linear and
Nε (bε ) = O(bε2 ) as kbε kH 0 → 0.
Consider equation for unknown hε :
hε = Lε (bε ) + Nε (hε ).
Can be solved in H 0 by contraction mapping; hε = bε is unique
small solution.
Want to show that for ε small, there exists s(ε) > 0 for which
contraction mapping method works in H s(ε) . If so, solution in H s(ε)
must coincide with solution in H 0 . (Pick one good ε.)
Key: linearization bε 7→ Lε (bε ) is smoothing, albeit with very bad
constant as ε → 0.
M. Christ, UCB Shuanglin Shao, IMA
Extremals For The Tomas-Stein Inequality
Problem and Main Results
Background/Remarks/Motivations/Open Questions
Existence of extremals — Proof
Properties of Extremals
Regularity
Complex-valued extremals
Local Maxima
The Complex Case
If f ∈ L2 (S 2 ) is a complex-valued extremal, then F = |f | is
extremal and
f = e ix·ξ |f | almost everywhere, for some ξ ∈ R3 .
Write f = e iϕ F .
|f σ ∗ f σ| = F σ ∗ F σ a.e. on B(0, 2) ⊂ R3 , so
f σ ∗ f σ(x) = e iψ(x) F σ ∗ F σ a.e. on B(0, 2).
Thus
e i[ϕ(x)+ϕ(x)] = e iψ(x+x
0)
a.e.
Want to show that e iψ(x) = ce ix·ξ . Then trivially same for ϕ.
M. Christ, UCB Shuanglin Shao, IMA
Extremals For The Tomas-Stein Inequality
Problem and Main Results
Background/Remarks/Motivations/Open Questions
Existence of extremals — Proof
Properties of Extremals
Regularity
Complex-valued extremals
Local Maxima
An S 2 additive quadruple is ~x = (x1 , x2 , x3 , x4 ) ∈ (S 2 )4 :
x1 + x2 = x3 + x4
e i[ϕ(x1 )+ϕ(x2 )] = e i[ϕ(x3 )+ϕ(x4 )] .
Given: a.e. quadruple with x1 + x2 = x3 + x4 is additive.
In same way, define R3 additive quadruples for ψ.
Easy: If ψ is additive a.e., then e iψ = ce ix·ξ for some c, ξ.
Device: If zj ∈ R3 , |zj | < 2 and z1 + z2 = z3 + z4 then there
exist (a.e.) two additive quadruples ~x , x~0 in (S 2 )4 satisfying
xj + xj0 = zj for all j ∈ {1, 2, 3, 4}.
Algebra ⇒ e i[ψ(z1 )+ψ(z2 )] = e i[ψ(z3 )+ψ(z4 )] .
M. Christ, UCB Shuanglin Shao, IMA
Extremals For The Tomas-Stein Inequality
Problem and Main Results
Background/Remarks/Motivations/Open Questions
Existence of extremals — Proof
Properties of Extremals
Regularity
Complex-valued extremals
Local Maxima
The last theorem: If {fν } is a complex-valued extremizing sequence
then there exists {ξν } ⊂ R3 such that {e −ix·ξ fν (x)} is precompact
in L2 (S 2 ).
This is proved by the above method, replacing each “a.e.” with
a quantitative refinement.
M. Christ, UCB Shuanglin Shao, IMA
Extremals For The Tomas-Stein Inequality
Problem and Main Results
Background/Remarks/Motivations/Open Questions
Existence of extremals — Proof
Properties of Extremals
Regularity
Complex-valued extremals
Local Maxima
Constants are Local Maxima
Constants are critical points.
Lemma. The second variation about 1 of our functional is
negative definite on the space of functions ⊥ 1 in L2 (S 2 ).
This boils down to a calculation of
ZZ
sup
g (x)g (y )|x − y |−1 dx dy
g ⊥1
S 2 ×S 2
with kg k2 = 1. We need a specific numerical bound.
This can be calculated explicitly using spherical harmonics and
Gegenbauer polynomials. The maximum is attained for
spherical harmonics of degree 2.
The required bound holds, with a factor of 3/5 to spare.
M. Christ, UCB Shuanglin Shao, IMA
Extremals For The Tomas-Stein Inequality
Problem and Main Results
Background/Remarks/Motivations/Open Questions
Existence of extremals — Proof
Properties of Extremals
Regularity
Complex-valued extremals
Local Maxima
Failures
We tried to guess trial functions which beat constants.
Natural candidates: Certain linear combinations of constants,
and spherical harmonics of degree 2. One can do the
calculation. We got to at best about 80% of the value for
constants.
Symmetrization: We didn’t find any way to symmetrize,
beyond reflection x 7→ −x. Our functional is not monotone
under symmetrization with respect to planes.
M. Christ, UCB Shuanglin Shao, IMA
Extremals For The Tomas-Stein Inequality